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# When using the half-angle formulas for trigonometric functions of alpha/2, I determine the sign based on the quadrant in which alpha lies.Determine whether the statement makes sense or does not make sense, and explain your reasoning. # When using the half-angle formulas for trigonometric functions of alpha/2, I determine the sign based on the quadrant in which alpha lies.Determine whether the statement makes sense or does not make sense, and explain your reasoning.

Question
Trigonometric Functions asked 2020-12-16
When using the half-angle formulas for trigonometric functions of $$\displaystyle\frac{\alpha}{{2}}$$, I determine the sign based on the quadrant in which $$\displaystyle\alpha$$ lies.Determine whether the statement makes sense or does not make sense, and explain your reasoning.

## Answers (1) 2020-12-17
For example:
The trigonometric function is given as, $$\displaystyle{\cos{{\left({112.5}^{\circ}\right)}}}.$$
$$\displaystyle{\cos{{\left({112.5}^{\circ}\right)}}}={\cos{{\left(\frac{{225}^{\circ}}{{2}}\right)}}}$$
Here, $$\displaystyle\alpha={225}^{\circ}.$$
Apply the half angle formula for the given trigonometric function, $$\displaystyle{\cos{{\left({112.5}^{\circ}\right)}}}$$ in which the angle $$\displaystyle{112.5}^{\circ}$$ lies in the quadrant (II) where sine and cosecant is positive. We have to determine the value of $$\displaystyle{\cos{{\left({112.5}^{\circ}\right)}}}$$ which is negative in second quadrant.
Therefore in half angle formula , negative sign has to be put.
$$\displaystyle{\cos{{\left(\frac{\alpha}{{2}}\right)}}}=-\sqrt{{\frac{{{1}+{\cos{\alpha}}}}{{2}}}}$$
The value of $$\displaystyle{\cos{{\left({112.5}^{\circ}\right)}}}={\cos{{\left(\frac{{225}^{\circ}}{{2}}\right)}}}{i}{s},$$
$$\displaystyle{\cos{{\left(\frac{{225}^{\circ}}{{2}}\right)}}}=-\sqrt{{\frac{{{1}+{\cos{{225}}}^{\circ}}}{{2}}}}$$
$$\displaystyle=\sqrt{{\frac{{{1}+{\left(-\frac{\sqrt{{2}}}{{2}}\right)}}}{{2}}}}$$
$$\displaystyle=-\sqrt{{{\left(\frac{{{2}-\sqrt{{2}}}}{{2}}\right)}}}$$

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